Choose Snapping Point?
This is the workflow I've found for doing this repetitively in Object mode.
The setup:

Pivot set to '3D Cursor'
Snap-magnet: 'Off'
Snap set to 'Vertex' and 'Center'. Because of the 'Pivot' setting, 'Center'  refers to the location of the 3D Cursor at the start of the move.
(The active tool is 'Tweak'.. it always is, for me)

The moves, for each snap:

Shift-Right  drag the cursor. Once you're in charge of the cursor, you can release the Shift.
Still Right-dragging, hold Ctrl to snap the cursor to the desired 'From' vertex. You have now established the snap 'From' point.
Now Left-drag the 'From' object to the 'To' object, with Ctrl, hovering over the target 'To' vertex. The 'From' point will snap to it.